Discussion
Let W be the subspace of \(\mathbf{R}^{4}\) spanned by the vectors \([1\ 2\ 0\ 0]^{t}\), \([0\ 1\ 2\ 0]^{t}\) and \([1\ 1\ -2\ 0]^{t}\). The dimension of the quotient space \(\mathbf{R}^{4}/W\) is :
Answer: B
The quotient \(\mathbf{R}^{4}/W\) has dimension 2.
Check the spanning vectors for dependence:
\[ [1\ 2\ 0\ 0] - [0\ 1\ 2\ 0] = [1\ 1\ -2\ 0] \]
So the third vector is the difference of the first two, and the first two are clearly independent. Hence dim W = 2 and
\[ \dim \mathbf{R}^{4}/W = 4 - 2 = 2 \]
Assuming the three vectors independent gives 1; counting only the zero fourth coordinate gives 3.
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